This implies that sin2⁡x-sin2⁡y≥0superscript2xsuperscript2y0\sin^{2}x-\sin^{2}y\geq 0 if and only if
cos2⁡y-cos2⁡x≥0superscript2ysuperscript2x0\cos^{2}y-\cos^{2}x\geq 0, which is equivalent to stating
that sin2⁡x≥sin2⁡ysuperscript2xsuperscript2y\sin^{2}x\geq\sin^{2}y if and only if cos2⁡x≤cos2⁡ysuperscript2xsuperscript2y\cos^{2}x\leq\cos^{2}y. Taking square roots, we conclude that
sin⁡x≤sin⁡yxy\sin x\leq\sin y if and only if
cos⁡x≥cos⁡yxy\cos x\geq\cos y.